Joints in graphs

In 1969 Erdős proved that if r⩾2r⩾2 and n>n0(r)n>n0(r), every graph G of order n   and e(G)>tr(n)e(G)>tr(n) has an edge that is contained in at least nr-1/(10r)6rnr-1/(10r)6r cliques of order (r+1)(r+1). In this note we improve this bound to nr-1/rr+5nr-1/rr+5. We also prove a corresponding stability result.